Ice-melt rates by steam condensation duringexplosive subglacial eruptionsD. C. Woodcock1, J. S. Gilbert1, and S. J. Lane1
1Lancaster Environment Centre, Lancaster University, Lancaster, UK
Abstract Subglacial volcanism melts cavities in the overlying ice. These cavities may be flooded withmeltwater or they may be fully or partially drained. We quantify, for the first time, heat transfer rates bycondensation of steam on the walls and roof of a fully or partially drained subglacial eruption cavity. Ourcalculations indicate that heat fluxes of up to 1MWm�2 may be obtained when the bulk vapor in the cavityis in free convection. This is considerably smaller than heat fluxes inferred from ice penetration rates inrecent subglacial eruptions. Forcing of the convection by momentum transfer from an eruption jet may allowheat fluxes of up to 2MWm�2, consistent with values inferred for the Gjálp 1996 subglacial eruption.Vapor-dominated cavities in which vapor-liquid equilibrium is maintained have thermal dynamic responsesthat are an order of magnitude faster than the equivalent flooded cavities.
1. Introduction
Subglacial volcanism melts cavities in the overlying ice. In most cases the resulting meltwater drainsfrom the eruption site to be released at the ice margin as a jökulhlaup or to be stored in a subglacial lake[Gudmundsson et al., 2004; Magnússon et al., 2012]. For temperate (wet-based) glaciers, the extent ofdrainage of the resulting meltwater appears to be determined by subglacial hydrology rather than byvolume changes occurring during magma-ice interaction [Höskuldsson and Sparks, 1997]. Thus, a cavitymay be liquid filled, fully drained, or partly drained. Figure 1 shows a case where a subglacial eruption cavityis partly drained.
Heat transfer during subglacial eruptions has been considered by Höskuldsson and Sparks [1997]; Wilsonand Head [2002, 2007], and Gudmundsson [2003]. Figure 11 in Gudmundsson [2003] shows a schematicillustration of a liquid-filled (flooded) cavity part way through a subglacial eruption. Woodcock et al. [2014]considered heat transfer within such a cavity by single-phase and boiling two-phase free convection. Tuffenet al. [2002] discussed the possibility of fully drained cavities in the context of subglacial rhyolitic eruptions.
We use the Gjálp eruption under the Vatnajökull ice cap, Iceland in October 1996 as a benchmark for ourcalculations because this is one of the best documented examples of a subglacial explosive eruption[Gudmundsson et al., 1997, 2004]. This eruption penetrated a thickness of 500–600m of ice in around 30 h afterthe start of the eruption. Ice-melting rates of up to 0.5 km3 per day were inferred by repeated observationsof depressions that developed in the ice surface above the eruption site and by changes in the volume of waterin the subglacial lake Grimsvötn.
1.1. Scope of Paper
If an ice cavity is fully or partially drained, rather than flooded, heat may be transferred frommagma to ice viasteam generation and condensation. For the fully drained case we envisage a predominantly steam-filledcavity within which a limited inventory of water repeatedly boils (by contact with magma) and condenses onthe roof and walls of the ice cavity. Although drained of meltwater, the cavity is deluged by an intense showerof meltwater and steam condensate from the cavity roof. There should thus be sufficient liquid wateravailable to promote phreatomagmatic fragmentation.
The possibility of heat transfer to ice by steam in subglacial eruption cavities has been discussed qualitativelyin the literature [Smellie, 2002; Tuffen et al., 2002]. In this paper we quantify (1) heat transfer rates fromsteam condensation on the ice cavity surface, using published heat transfer methods to estimate likelyheat fluxes, (2) the thermal dynamic response of a vapor-dominated cavity to a change in heat input, and(3) the conditions under which a fully or partially drained subglacial cavity may exist.
Citation:Woodcock, D. C., J. S. Gilbert, andS. J. Lane (2015), Ice-melt rates by steamcondensation during explosivesubglacial eruptions, J. Geophys.Res. Solid Earth, 120, 864–878,doi:10.1002/2014JB011619.
Received 18 SEP 2014Accepted 11 JAN 2015Accepted article online 14 JAN 2015Published online 11 FEB 2015
Figure 2a shows part of the sloping wall ofthe ice cavity and summarizes the variousaspects of heat transfer involved. Heat istransferred from the cavity bulk vapor to theice surface through two thermal resistancesin series: a gas boundary layer and a liquidfilm of condensate and meltwater. The gasboundary layer resistance is principally dueto the presence of noncondensable gases,which, in the context of this study, includesany gaseous component that does notcondense or dissolve in the liquid film. Thenature of this resistance is as follows. Flowfrom the bulk vapor to the liquid-vaporinterface transports both steam, whichcondenses, and noncondensable gases,which accumulate at the interface (Figure 2b).At steady state, an equivalent diffusionalcounterflow of noncondensable gases awayfrom the interface is driven by an elevatedpartial pressure of noncondensable gases atthe interface. Since the total pressureremains constant, the partial pressure ofsteam at the interface is reduced. Thecondensing temperature of steam at theinterface is thus reduced, which in turnreduces the temperature driving force forheat transfer across the liquid film [Collierand Thome, 1994].
The effect of noncondensable gases oncondensing heat transfer is of considerablepractical importance in engineering;particularly in the nuclear industry, wheresteam condensation in the presence ofnoncondensable gases is an important
Figure 1. Schematic diagram of a partly drained subglacial eruption cavity that drains meltwater into a subglacial lake.
Figure 2. (a) Summary of various aspects of heat transfer on the slopingwall of a drained but steam-filled ice cavity. Heat is transferred from thecavity bulk vapor to the ice surface through two thermal resistancesin series: a gas boundary layer and a liquid film. (b) The influence ofnoncondensables on interfacial resistance. The partial pressures ofvapor and of noncondensables in the bulk and their variation throughthe gas-vapor film are shown diagrammatically (the ice surface isvertical for ease of comprehension). The corresponding variation invapor saturation temperature is also shown. The reduction in saturationtemperature at the interface causes a reduction in heat transferthrough the liquid film compared to the case where noncondensablesare absent.
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mechanism for removing heat from a reactor containment vessel in the case of certain accident scenarios[Anderson et al., 1998; De la Rosa et al., 2009]. A reactor containment vessel is typically 20–30m in diameter withthe bulk flow in turbulent free convection during steam condensation once the initial reactor blowdown iscomplete [Kim and Corradini, 1990; Kim et al., 2009]. The extensive theoretical and experimental work devotedto quantifying steam condensation on the cooled surfaces of reactor containment vessels is thus of relevance tosteam condensation on the roof and walls of a subglacial cavity.
2. Method
In this section we summarize the development of a model to estimate heat transfer rates during steamcondensation. We review results for the condensation of pure vapors on vertical cooled surfaces, theextension to downward facing inclined cooled surfaces, and the modification required for simultaneouscondensation and melting. We then consider condensation in the presence of noncondensable gases.
2.1. Condensation of a Pure Vapor on Vertical Cooled Surfaces
The analysis of condensation heat transfer can be traced back to Nusselt’s 1916 paper [Rose, 1998], where hederived an equation for the heat transfer coefficient during condensation of a pure vapor on a vertical cooledsurface. Nusselt’s analysis was restricted to laminar flow in the condensate film. Subsequent work hasconsidered condensation at higher film Reynolds numbers. The film Reynolds number, Re, is defined as
Re ¼ 4G=μl (1)
where G is the mass flow rate of condensate per unit width of the film and μl is the condensate viscosity.The film surface is smooth for Re < 30. Above this value, ripples and waves form on the surface of the film:in this “wavy laminar” regime heat transfer increases, principally by reducing the average film thicknessbut also by increasing the area available for heat transfer. For Re> 1600, a fully turbulent, wavy film isdeveloped [Incropera and DeWitt, 1996]. Marto [1998] presents a correlation of experimental results for therange 10< Re < 31,000:
Nu ¼ 1:33Re�1=3 þ 9:56� 10�6Re0:89Pr 0:94 þ 0:082 (2)
where Nu=U (μl2/ρl
2g)1/3/kl, U is the heat transfer coefficient, ρl and kl are the liquid density and thermalconductivity respectively, g is the gravitational acceleration, and Pr is the liquid Prandtl number. The first term,which dominates at low Re, is similar to the classical form of the Nusselt equation [Incropera and DeWitt, 1996]:
Nu ¼ 1:47Re�1=3 (3)
2.2. Condensation of a Pure Vapor on Downward Facing Inclined Cooled Surfaces
Nusselt’s original analysis was restricted to vertical surfaces. In a subsequent paper he considered condensationon the outside of a horizontal tube by proposing that, for a surface inclined at θ to the horizontal, thegravitational acceleration is “diluted” by a factor of sin θ [Rose, 1998]. For inclined surfaces, this has proved to bean excellent modification, at least for large θ, even on downward facing surfaces, where the liquid film issubjected to a Rayleigh-Taylor type instability [Gerstmann and Griffith, 1967; Piriz et al., 2006].
Gerstmann and Griffith [1967] studied heat transfer for condensation on the underside of inclined surfaces,using Freon 113 as the condensing fluid and taking care to eliminate noncondensable gases. For inclinationsgreater than 15° from the horizontal their experimental results were within 10% of those predicted by themodified Nusselt equation. This result was corroborated by Chung et al. [2005], who condensed steam ondownward facing plates as part of a wider study involving both steam and air.
2.3. Simultaneous Condensation and Melting of a Single Component
Most work on condensing heat transfer in the literature is concerned with condensation on cooled surfaces.For heat transfer within an ice cavity the latent heat of condensation melts the ice surface: the resultantliquid film thus comprises a mix of steam condensate and ice meltwater. Nusselt’s analysis can be readilymodified to allow for the thicker film that results. The heat transfer coefficient should bemultiplied by a factorof (1 + λv/λf )
�1/4, where λv and λf are the latent heats of vaporization and fusion, respectively [Eckhardt, 1968].For steam at 0.1MPa this factor is 0.6 (0.64 at 4MPa). The condensation of steam on ice in the presence of airwas studied experimentally by Yen et al. [1973]. In their air-free experiments they found that simultaneous
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condensation and melting effectively halved the heat transfer coefficient, compared with condensation on asurface cooled to 0°C.
2.4. Condensation in the Presence of Noncondensable Gases
Sections 2.1–2.3 developed a model for the heat transfer within the combined condensate and meltwaterfilm on the ice surface. This section reviews models for the heat and mass transfer through the gas boundarylayer that develops if noncondensable components are present in the bulk gas.
Analytical solutions of the boundary layer equations were achieved for simple geometries in the late 1960s[Minkowycz and Sparrow, 1966; Sparrow et al., 1967]. However, the solutions were too complex for practicalapplication at the time; thus, industry relied on empirical correlations for the pragmatic solution of problemsinvolving condensation in the presence of noncondensable gases [Herranz et al., 1998].
In the early 1990s, an alternative theoretical approach was developed that makes use of the analogy betweenheat and mass transfer in order to evaluate the mass transfer coefficients within the diffusional vapor-gasfilm. These heat and mass transfer analogy (HMTA) models or “diffusion layer models” [Herranz et al., 1998]became widely adopted within the nuclear and process industries [De la Rosa et al., 2009]. The developmentof the HMTA model is documented by Collier and Thome [1994]. We use the HMTA model, together withequation (2) with g replaced by g sin θ for the liquid film resistance and the modification for simultaneouscondensation and melting (section 2.3) to explore the effect of noncondensable gases’ concentration andpressure on condensation in subglacial cavities.
The HMTA model requires evaluation of the heat transfer coefficient in the gas phase.
The intensity of free convection may be characterized by the Rayleigh number, Ra, defined as
Ra ¼ gaΔTd3
κv(4)
where g is the gravitational acceleration, ΔT is the temperature difference, d is the vertical dimension ofthe convective motion, and α, κ, and ν are the coefficients of thermal expansion, thermal diffusivity, andkinematic viscosity, respectively, of the fluid [Turner, 1973]. For Rayleigh numbers appropriate to convectionin subglacial cavities, the heat transfer coefficient over a smooth surface inclined at an angle θ to thehorizontal may be obtained from the Nusselt number, Nu, given by [Raithby and Hollands, 1998]
Nu ¼ max 0:13 Ra sin θð Þ1=3; 0:14 Ra cos θð Þ1=3n o
(5)
where Nu is defined as Ud/kf, with free convection heat transfer coefficient U and fluid thermal conductivity kf..
The resulting heat transfer coefficient needs to be enhanced by a factor to allow for the roughness of the liquidfilm surface. Kim et al. [2009] carried out awide-ranging study of condensation in the presence of noncondensablegases for pressures in the range 0.4–2.0MPa. We have used their data, together with the HMTA model, todetermine an appropriate enhancement factor (EF). This appears to be independent of the mole fraction ofnoncondensable gases but depends on pressure. The resulting equation for the enhancement factor is
EF ¼ 2:6 þ 0:4P (6)
where P is the pressure in MPa.
The study by Kim et al. [2009] was restricted to vertical surfaces. Huhtiniemi and Corradini [1993] studiedcondensation heat transfer in the presence of noncondensable gases at atmospheric pressure for a range ofdownward facing surface orientations. They found little effect of orientation on heat transfer coefficient,suggesting that the enhancement factor might be independent of orientation. Furthermore, their data canbe reproduced well by our HMTA model with an enhancement factor of 2.6 (appropriate to 0.1MPa).
3. Results3.1. General Results for the HMTA Model
Figure 3 shows overall heat transfer coefficients versus mole fraction of noncondensable gases in the bulkvapor, for various total cavity pressures and liquid film lengths. In all cases the overall heat transfer coefficientdecreases with an increase in the mole fraction of noncondensable gases. Similar trends in overall heat transfer
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coefficient with noncondensable gas molefraction have been reported in the literature.Chung et al. [2005] performed a seriesof steam condensation experiments atatmospheric pressure on a water-cooled flatplate, while varying the plate inclination andthe proportion of air in steam. Their resultsshowed a systematic reduction of overallheat transfer coefficient with increasingproportion of air. Heat transfer coefficientsare relatively insensitive to liquid film lengthexcept at very low (<0.01) mole fraction ofnoncondensables.
3.2. Application to Subglacial Cavities
The general behavior of the HMTA modelsuggests that heat transfer coefficientsdepend mainly on bulk noncondensables’mole fraction and cavity pressure. Thissection attempts to estimate the likely rangeof heat transfer coefficients that might beobtained in drained subglacial cavities.3.2.1. The Effect of NoncondensableGasesHöskuldsson and Sparks [1997] suggestedthat noncondensable gases could besourced from air released from ice onmelting and from insoluble componentsof volcanic gases. Air in glacial ice
originates as air trapped within firn [Martinerie et al., 1992]. Firn is permeable to air, so during compactionany excess air can be vented to the surface via a network of interconnecting channels. At a “critical” depthof around 50–100m this connection to the surface becomes broken and air, at atmospheric pressure,becomes trapped as air bubbles. The amount of air released on melting can be estimated, given that firnvoidage at the critical depth is around 0.1 [Martinerie et al., 1992]. Thus, at the critical depth, 1m3 of firncomprises 0.9m3 of pure ice (around 828 kg) and 0.1m3 of air at atmospheric pressure (around 0.13 kg).Hence, around 0.16 kg of air per ton (103 kg) of ice will be released on melting.
Sigvaldson and Elisson [1968] present chemical analyses of volcanic gas from the 1965 Surtsey eruption.A typical analysis, for a case where there is minimal contamination by ambient air, is presented in Table 1.Both HCl and SO2 are highly soluble in water and can be expected to dissolve completely in meltwater, whileH2 and CO are substantially insoluble in water. CO2 is partly soluble in water, dependent on its partial pressureand water temperature.
To establish the likely concentrations of noncondensable gases, we assume for illustrative purposes a subglacialeruption of basaltic magma in a drained cavity filled with saturated steam at 1.5MPa. If the magma cools froman eruption temperature of around 1200°C to a glass at 200°C, the heat released is 1200kJ per kg magma,assuming an average glass specific heat capacity of 1.2 kJ kg�1 K�1 [Höskuldsson and Sparks, 1997]. If the heat
Table 1. Volcanic Gas Composition From 1965 Surtsey Eruption
aTable 1 (average for 21 February) in Sigvaldson and Elisson [1968].
Figure 3. Heat transfer coefficients from bulk cavity fluid to the icewall (inclined at 15°) versus mole fraction of noncondensable gasesin the bulk vapor. (a) The effect of cavity pressure, for a liquid filmlength of 0.5m. (b) The effect of liquid film length at a cavity pressureof 0.5 MPa.
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released melts ice and heats meltwater to 20°C, the amount of ice melted is thus 1200/(335+4.2 × 20) oraround 3 kg per kgmagma, where the latent heat of fusion of ice is 335 kJ kg�1 and the specific heat capacity ofwater is 4.2 kJ kg�1 K�1 [Rogers and Mayhew, 1980]. For each ton of magma, the size of the cavity produced isthus around 3.3m3 and the amount of air released by melting the ice is around 0.5 kg.
Mid-ocean ridge basalt and ocean island basalt magmas contain less than 1% weight of volatiles on eruption[Wallace and Anderson, 2000]. One ton of magma containing 1% of gas with the composition presented inTable 1 would release 0.11 kg of H2 and CO together with 1.25 kg of CO2 as potential noncondensable gases.CO2 solubility is limited at 200°C (the temperature of the steam-water interface in the cavity), given therelatively low partial pressure of CO2 in the cavity. The total amount of noncondensable gases is thusaround 1.5–2 kg per ton of magma. If the cavity produced by the eruption is drained, it will be filled withsaturated steam at 200°C with a density of 8.1 kgm�3 [Rogers and Mayhew, 1980]. A cavity of 3.3m3 thuscontains 27 kg steam. The corresponding level of noncondensable gases in the steam is thus around 5weight%;corresponding to a mole fraction of 0.02.
The calculation above assumes that the cavity produced is empty; in practice some of the cavity (around10–20%) will be occupied by the volcanic edifice. Furthermore, if the cavity is significantly underpressured(compared with glaciostatic conditions) then the cavity will progressively collapse by ductile ice flow [Tuffen,2007]. Both effects will reduce the space available for the noncondensable gases and thus increase theirconcentration within the cavity. The Katla 1918 eruption penetrated the overlying ice in 2 h [Gudmundsson,2005], and there was probably little ductile movement in the overlying ice during the subglacial stage of theeruption. For the Gjálp 1996 eruption, the overlying ice was penetrated in 30 h [Gudmundsson et al., 2004],allowing sufficient time for ductile ice flow to reduce cavity size (depressions were observed in the ice surfaceabove the eruption site). It seems likely that, in the case of the Gjálp 1996 eruption, the concentration ofnoncondensable gases in the cavity may have been considerably greater than 5 weight % unless they wereremoved from the cavity.
A possible way of removing noncondensable gases from the cavity might be by venting them to the surface upa crevasse in the ice. Gudmundsson et al. [2004] noticed the presence of en echelon fractures on the ice surfaceabove the Gjálp 1996 site before the eruption became subaerial. They suggested that the fractures may bethe surface expression of a basal crevasse that was produced by tensional stresses associated with dykeintrusion at the start of the subglacial eruption and that fractures might be of the order of 1m wide.
If a fracture in the ice exists from cavity to surface, then amixture of steam and noncondensable gasesmay ventfrom the cavity. During flow up in the fracture, steam would progressively condense andmelt back the walls ofthe fracture, thus mitigating the tendency for ductile ice flow to close the fracture (note that both creeprates and meltback rates increase downward). If the fracture is sufficiently wide, the combined liquid flow ofcondensate and meltwater will reflux back down the fracture. Application of the “flooding equation” [Kay andNedderman, 1985] suggests that liquid can reflux if the fracture is wider than 0.5m. Note that the floodingequation is applicable to vertical circular pipes and may not be appropriate for a channel with parallel walls.At the surface, only the noncondensable gases would be vented; these would not be visible from the air;however, it might be possible to detect elevated levels of noncondensable gases above the fractures.3.2.2. Implications for Heat Fluxes in Drained Subglacial CavitiesIn section 4.1 we suggest that liquid film thickness is unlikely to increase for film lengths above 0.5m.The discussion in section 3.2.1 suggests that the mole fraction of noncondensables is likely to be at least0.02 but that it is difficult to set an upper limit. For a liquid film length of 0.5m and a mole fraction ofnoncondensables of 0.02, Figure 3 indicates that heat transfer coefficients range from 1 to 4 kWm�2 K�1 overthe pressure range 0.1 to 4MPa. The corresponding heat fluxes are thus 0.1 to 1MWm�2.
A condensing heat flux of 1MWm�2 will melt ice with a vertical penetration rate of around 12mh�1 ifmeltwater leaves the ice cavity surface at 0°C. If the meltwater temperature is 100°C the corresponding rate is5mh�1. Ice penetration rates of 16 and 50mh�1 were inferred for the Gjálp 1996 eruption [Gudmundsson et al.,2004] and the 2010 Eyjafjallajökull eruption [Magnússon et al., 2012], respectively. Thus, heat fluxes from steamcondensation appear to be considerably smaller than those inferred from recent subglacial eruptions. Wespeculate that, if a steam layer undergoing free convection developed during the Gjálp 1996 subglacialeruption, it was only present for part of the eruption and that other, more efficient, ice penetrationmechanismswere present at other times. We consider possible ways of enhancing the steam condensation in section 3.2.3.
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3.2.3. Forced Convection inSteam-Dominated CavitiesThe results presented in section 3.2.2assume that heat transfer in the gas phaseis by free convection; in some instances,convection within the gas phase may beenhanced by momentum transfer froman eruption jet.
Figure 4 shows the variation of overall heattransfer coefficient (OHTC) with gas filmheat transfer coefficient hf. Figure 4 wasdeveloped from a modified version of theHMTA model where hf can be variedindependently rather than calculated fromequation (5). At 2MPa, values of hf of100Wm�2 K�1 are typical of free convection.As hf increases the heat transfer resistance
of the gas-vapor film decreases, and the OHTC is increasingly determined by the heat transfer of the liquidfilm. In this case a factor of 2–3 increase in the OHTC can be obtained, depending on the concentration ofnoncondensables present. Forced convection within a steam-dominated cavity may thus melt ice with an icepenetration rate to match that inferred for the Gjálp 1996 eruption.
It is possible that the overall heat transfer coefficient could be increased further by impingement of themagmatic eruption jet onto the ice surface, causing local thinning of the liquid film. However, this wouldrequire the jet to be optimally placed relative to the ice surface, which is unlikely. Furthermore, anyenhancement in melting rate would be restricted to the immediate locality around the impingement zone;the rest of the cavity roof would remain largely unaffected.
3.3. Thermal Dynamics of Steam-Dominated Cavities
Themodel presented in sections 2 and 3 so far assumes steady state heat transfer, where steam condensationrates are balanced by steam generation rates and cavity pressure is constant. In this section we consider thethermal dynamics of a steam-dominated cavity in response to changes in steam generation rate. Suchchanges may be caused by changes in magma input or by changes in the efficiency of heat transfer frommagma to water, for example, by changes in the degree of magma fragmentation. One might expect that asteam-dominated cavity would have a small thermal capacity and thus a dynamic response that is orders ofmagnitude faster than the equivalent liquid-filled cavity. Such rapid changes in cavity conditions may haveimplications for ice-melting rates and thus for the development of the associated eruption hazards. In thissection we show that, if vapor-liquid equilibrium is maintained, the effective thermal capacity is dominatedby latent heat changes that are much larger than sensible heat changes.
Consider a cavity at steady state that is subjected to a sudden increase in steam generation rate. Cavitypressure will increase, along with cavity temperature if thermal equilibrium between steam and liquid wateris maintained. The temperature of the magma provides the ultimate upper bound on cavity temperature;however, we restrict our discussion to cavity temperatures below the critical temperature of water. Under thisrestriction, cavity pressure may increase up to the critical pressure of 22.1MPa. The glaciostatic head of iceequivalent to the critical pressure is around 2500m; thus, the ice sheet surrounding a subglacial eruptionmight be expected to lift or to fracture in most if not all terrestrial subglacial eruptions.
For illustrative purposes, we estimate the time taken for cavity pressure to double in response to a doublingof steam generation rate. For expediency, we consider a triangular cavity of height H and base 2βH thatdevelops over a linear eruption fissure. The surface area for steam condensation per unit length of fissure isthus 2H (1 + β2)1/2. For a condensation heat transfer coefficient U and a steady state cavity temperature Tso,the condensing heat load Q (equal to the steam generation heat load in steady state) is given by
Q ¼ 2UT soH 1þ β2� �1=2
(7)
Figure 4. Variation of overall heat transfer coefficient with gas filmheat transfer coefficient for a range of noncondensable mole fractions.Condensation and melting at 2MPa with a liquid film length of 0.5mon an ice cavity wall inclined at 15°.
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The steam generation heat load is then stepped up to 2Q. If the condensation heat transfer coefficient Uremains constant, the condensation heat load increases to QTs/Tso for a cavity temperature Ts that increaseswith time. The heat accumulation rate in the cavity, AC, is thus given by
AC ¼ Q 2� Ts=T soð Þ (8)
which is equal to the rate of change of cavity heat content with time.
The cavity heat content Hc, relative to liquid water at 0°C, is given by
Hc ¼ mlhl þmghg (9)
whereml andmg are the masses of liquid water (with enthalpy hl) and steam (with enthalpy hg) in the cavity.We consider the case where ml may be neglected but there is sufficient liquid water present to maintainvapor-liquid equilibrium as cavity pressure increases and liquid water is turned to steam. An upper limit onthe rate of change of cavity heat content with time is thus given by
dHc
dt¼ d mghg
� �dTs
� dTsdt
(10)
which may be expanded to
dHc
dt¼ dTs
dt� mg
dhgdTs
þ hgdmg
dTs
� �: (11)
The enthalpy of steam, hg, is almost independent of temperature over the range of interest (see Table 2);thus, equation (11) becomes
dHc
dt¼ hg
dTsdt
� dmg
dTs: (12)
For a cavity volume of βH2 per unit length of eruption fissure, the mass of steam in cavity is ρg βH2, where ρg is
the density of steam. Equation (12) thus becomes
dHc
dt¼ hg
dTsdt
�βH2 dρgdTs
: (13)
The effective thermal capacity per unit volume for a vapor-dominated cavity in which vapor equilibrium ismaintained is given by hg dρg/dTs. This is a latent heat term that is much larger than the sensible heat term ρgCpg in the case where vapor equilibrium is not maintained.
Combination of equations (7), (8), and (13) yields, after rearrangement
dTsdt
¼ UT so 2� Ts=T soð Þhg
� 2 1þ β2� �1=2
βH� dTs
dρg: (14)
The rate of increase of cavity temperature with time may be determined approximately by replacingdifferential terms by finite differences. The finite difference ΔTs/Δρg is evaluated over a range of cavitytemperatures in Table 2.
Table 2. Evaluation of the Finite Difference ΔTs /Δρg (Equation (14)) Evaluated Over a Range of Cavity Pressuresa
For illustration, consider a cavity with a heightH of 50m, with β =2, and a condensing heat transfer coefficient Uof 2300Wm�2 K�1. For a cavity pressure increase from 2 to 4MPa, temperature increases at around 0.03 K s�1;the pressure increase thus takes around 20min. This estimate could be refined by considering alternative cavityshapes, the increase in condensing heat transfer coefficient with pressure, or variation in steam volume(perhaps produced by changes in water level in the cavity). The corresponding time for a liquid-dominatedcavity of the same size to respond to the same change in heat input is around 2h, around 6 times slower.
3.4. Conditions for the Existence of a Partly Flooded Subglacial Eruption Cavity
Figure 1 shows a case where a subglacial eruption cavity is partly drained. Here meltwater discharges fromthe cavity via a meltwater pathway in the ice into a subglacial lake. Under some conditions, a steam-liquidwater interface can be present in the cavity, with the equilibrium level L of the interface determined by
Psteam þ Lρlg ¼ PD þ ΔPf (15)
where Psteam is the steam pressure in the cavity, ρl is the density of liquid water, g is the gravitationalacceleration, PD is the static pressure at the point where the meltwater enters the subglacial lake and ΔPf isthe frictional pressure loss in the conduit. Variation of the interface level in the cavity has the potential tovary meltwater flow rates and to vary the meltwater production rates if heat transfer rates above and belowthe interface are significantly different.
We examine in more detail the conditions under which a steam-liquid water interface may exist within asubglacial cavity. Figure 5 shows a cross section along the meltwater drainage route from the center lineof a cavity of height H to a subglacial lake where the meltwater accumulates. The extent of drainage of thecavity may be quantified by L/H, with the depth of liquid water L in the cavity determined by consideringthe static pressure balance within the system.
At the top of the cavity (point B in Figure 5), the pressure PB is determined by the thickness of ice and anyshear stress within the ice that reduces the glaciostatic pressure (this shear stress develops as a consequenceof progressive ductile collapse of the cavity roof as the cavity grows) [Gudmundsson et al., 2004]:
PB ¼ Gi � di � Hð Þρig� τ (16)
where Gi is the preeruption thickness (for the current eruption) of ice above bedrock, di is the depth of anydepression that develops in the ice surface and τ is the underpressure: the reduction in glaciostatic pressuredue to shear stress in the ice. The pressure at point C at the base of the cavity is thus
PC ¼ PB þ Lρlg (17)
if the density of steam is neglected. The pressure at point D at the entrance to the subglacial lake is thus
PD ¼ PC þ zρlg� ΔPf (18)
Figure 5. Schematic diagram of one half of a subglacial eruption cavity that drains meltwater into a subglacial lake, showingsymbols used in section 3.4. Volcanic edifice not shown.
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where ΔPf is the frictional pressure dropdue to flow of meltwater from C to Dand z is the elevation of C above D.Elimination of PB and PC from equations(16) to (18) and rearrangement yields
L ¼ PD þ τ þ ΔPfð Þρlg
� Gi � di � Hð Þρiρl
� z (19)
We consider equation (19) in the contextof the Gjálp 1996 subglacial eruption.Figure 14a in Gudmundsson et al. [2004]shows a cross section, part way throughthe subglacial eruption, along themeltwater path from the eruption site toGrimsvötn. This shows a thickness, h, of370m of ice where meltwater entersGrimsvötn. In general, the glaciostaticpressure is given by hρig, where ρi is thedensity of ice (911 kgm�3) and g is thegravitational acceleration (9.81ms�2).
The pressure PD, for this specific case, is thus 3.3MPa. The elevation difference between points C and D in Figure 5for this case is approximately 40m. Equation (19) can be used to estimate the ice thickness required for a steamlayer, of thickness (H � L), to exist at the top of a partly flooded subglacial cavity. Gudmundsson et al. [2004]suggested that the underpressure in the ice τ lies in the range 1–2MPa. The frictional pressure drop is likely tobe small compared to τ: we estimate a value for ΔPf of 0.1MPa in Appendix A.
For illustrative purposes we consider the case with τ equal to 1.5MPa. As cavity height H increases from 100mto 500m, the ice thickness Gi required for a 50m thick steam layer decreases from 549m to 510m. For a100m thick steam layer the corresponding ice thickness required decreases from 604m to 565m. An increasein τ of 0.5MPa increases the required ice thickness by 56m. Figure 14a in Gudmundsson et al. [2004] showsa preeruption ice thickness of around 600m. It is thus plausible that the Gjálp 1996 subglacial eruptioncavity was partly steam filled. In practice, large cavity heights may be infeasible if surface fractures in the iceextend through the cavity roof.
Equation (19) may be generalized by defining a “dimensionless steam depth”:
H� LGi
¼ Gi � di � Hð Þρi=ρl � τ=ρlgþ Hþ zGi
� PD þ ΔPfρlgGi
(20)
The second term on the right-hand side (RHS) is a “dimensionless resistance” to the flow of meltwater,comprising both backpressure (PD) and frictional drop ΔPf. The first term on the RHS may be considered as adimensionless driving force. A subglacial cavity is just flooded (full of liquid water) when L=H, i.e., when thedimensionless steam depth is zero. A subglacial cavity is just drained when L= 0, i.e., when the dimensionlesssteam depth is H/Gi. Negative values of dimensionless steam depth indicate that there is insufficientglaciostatic head above the top of the cavity to allow meltwater to drain, even when full of liquid water. Inpractice, negative values of dimensionless steam depth are infeasible: in this case τ would decrease to thepoint that meltwater could drain from the liquid-filled cavity.
Figure 6 shows a graph of dimensionless steam depth versus dimensionless driving force for two values ofdimensionless resistance. Data specific to a developing subglacial eruption may be plotted onto Figure 6.For illustration, two cases are plotted, using the Gjálp eruption data and a constant value of τ but withdifferent initial ice thicknesses. We assume here that di and ΔPf are constant, although they are expected toincrease and decrease respectively as the cavity develops. For an initial ice thickness of 500m the cavity isinitially flooded (dimensionless steam depth< 0) but develops a steam layer as cavity height increases. For an
Figure 6. Dimensionless steam depth versus dimensionless driving forcefor meltwater drainage during subglacial eruption cavity development.Dotted lines are lines of constant dimensionless resistance (DR) tomeltwater flow. The diagonal solid lines represent the development oftwo specific eruption cases (with constant DR), using Gjálp 1996 eruptiondata but for different initial ice thicknesses. Dashed lines join points ofconstant cavity height. Two additional solid lines identify the 50m and100m constant steam depth cases discussed in section 3.4.
Journal of Geophysical Research: Solid Earth 10.1002/2014JB011619
initial ice thickness of 600m the cavity is fully drained until the cavity height is around 100m and thenpartly drained for the remainder of its development, with the steam layer increasing in thickness to 140m.Development of a steam layer in a cavity is thus favored by thick ice that provides sufficient glaciostaticpressure to balance the resistance to meltwater drainage.
4. Discussion
In this section we discuss controls on liquid film length and thickness during steam condensation, comparesteam condensation heat fluxes with likely heat fluxes from other heat transfer mechanisms, and consider theeffect of tephra in the system.
4.1. Controls on Liquid Film Length and Thickness
On a vertical surface, the liquid film length can extend down the full height of the surface. Irregularities in thesloping surface of the ice cavity may limit the length of a liquid film, with shedding of the film if a localdiscontinuity of slope is sufficiently abrupt. We anticipate that the cavity surface may be fractured bymagmatic and phreatomagmatic explosive activity; in this case liquid film length will depend on the degreeto which the cavity surface is fractured. Fracturing will also increase the ice surface area available for heattransfer. We are not aware of information in the literature on fracture spacing due to explosions under ice.
There is some evidence that the film thickness may be self-limiting. On downward facing inclined surfaces,Gerstmann and Griffith [1967] observed that the condensate film developed into a series of longitudinalridges and troughs and that condensate rained from the ridges. They envisaged a steady state in whichmuchof the condensation took place at the thin film in the troughs and that condensate flow had a lateralcomponent into the ridges, where it was removed from the surface by droplet shedding. Under this scenario,a condensate film may attain a constant average thickness, rather than increasing along its length.
Gerstmann and Griffith [1967] observed that the location of the transition between the initially smooth filmand the alternating ridges and troughs moved up the downward facing inclined surface as heat flux, andthus condensation rate, increased. In all cases they observed that the transition occurred within the length ofthe surface in their apparatus (approximately 0.5m). For the case of simultaneous condensation and melting,in which the liquid flux in the film is increased by an order of magnitude compared with the simple cooledsurface case, one might expect an earlier transition to ridges and troughs.
Gerstmann and Griffith’s work was carried out with Freon 113, rather than water. Anderson et al. [1998]observed similar behavior for the condensation of steam on the underside of a cooled curved surface thatincreased in inclination from horizontal to 15° over a 1.8m length. We tentatively conclude that, forsimultaneous condensation and melting on a downward facing inclined surface, a maximum average filmthickness might be attained after 0.5m.
4.2. Comparison of Steam Condensation With Other Heat Transfer Mechanisms
In section 3.2.2 we show that, in vapor-dominated cavities, heat fluxes of 0.1–1MWm�2 can be obtained bysteam condensation where the bulk steam undergoes free convection and we anticipate heat fluxes up to2MWm�2 for forced convection. For liquid-dominated cavities, 0.1 to 1MWm�2 may be achieved bysingle-phase free convection, with heat fluxes of 3–5MWm�2 estimated for two-phase free convection[Woodcock et al., 2014]. Thus, steam condensation heat fluxes are similar to those likely to be achieved bysingle-phase convection in liquid-filled cavities but smaller than heat fluxes estimated for two-phase freeconvection. In a partly liquid-filled cavity undergoing two-phase free convection, sideways melting rates maybe considerably greater than upward melting rates. This is similar to the findings of Höskuldsson and Sparks[1997], who produced pancake-shaped cavities when an insulating layer of air developed at the top ofcavities in their experiments to melt ice by single-phase convection of liquid water.
An alternative ice-melting mechanism, in which heat is transferred directly frommagma to ice, was proposedby Wilson and Head [2002] for basaltic magmas. They proposed that a depressurized (i.e., atmosphericpressure) subglacial cavity may be produced if a connection to the atmosphere can be established.Depressurization of the cavity initiates a Hawaiian-style lava fountain that “drills” through the overlying ice bypyroclast impact up to the unconstrained height of the fountain. Melting of the cavity then continues by
Journal of Geophysical Research: Solid Earth 10.1002/2014JB011619
radiative heat transfer from the lava fountain [Wilson and Head, 2002]. The heat fluxes in this scenario remainto be quantified.
4.3. The Effect of Tephra in the System
A vapor-dominated cavity will contain a “dusty gas” comprising tephra as well as steam. The model of steamcondensation heat transfer presented in this paper considers the tephra-free end-member and thus assessesthe contribution of steam condensation to ice-melting rates.
Replacing pure steam by a dusty gas with a higher density and specific heat capacity increases the rate oftransport of heat within the cavity, provided that convection velocities are not decreased by the higherdensity. Tephra particles may be carried up to the cavity roof with sufficient momentum to mechanicallyabrade the ice surface. In addition, the presence of particles enhances heat transfer coefficients within theliquid film on the ice surface [Ozbelge, 2001]. The presence of tephra may thus increase heat fluxes; however,the effect of tephra in the system remains to be quantified.
5. Conclusions
We have used published heat transfer calculation methods to estimate steam condensation heat fluxes onthe ice-melting surfaces within vapor-dominated subglacial cavities during explosive volcanic eruptions.The principal conclusions are as follows.
1. Heat fluxes of 0.1–1MWm�2 can be obtained by steam condensation in vapor-dominated cavities wherethe bulk steam undergoes free convection. These heat fluxes are similar to estimates of heat flux fromsingle-phase convection in liquid-filled cavities.
2. Forced convection of the bulk steam reduces the thermal resistance of the gas boundary layer and thusincreases the condensing heat flux to the point where heat transfer is determined by the resistance ofthe liquid film. In this case a maximum heat flux of 2MWm�2 may be obtained; this may produce icepenetration rates that match those inferred from recent subglacial eruptions.
3. Vapor-dominated cavities in which vapor-liquid equilibrium is maintained have thermal dynamicresponses that are an order of magnitude faster than the equivalent flooded cavities.
Appendix A: Frictional Pressure Drop in Subglacial Conduits
The frictional pressure drop ΔPf due to the flow of a fluid in a conduit of circular cross section with length Leand diameter D is given by:
ΔPf ¼ Cf � LeD
� ��0:5ρv2 (A1)
where ρ and v are the density and velocity of the fluid, respectively. Cf is the friction coefficient whichdepends on the Reynolds number of the flow and the relative roughness of the conduit wall [Massey, 1970].The flow rate of fluid q is as follows:
q ¼ πD2v=4 (A2)
For the Gjálp 1996 eruption considered in section 3.4 the flow rate was 0.5 km3 per day for the first two daysof the eruption [Gudmundsson et al., 2004] or 6000m3 s�1.
We expect the water velocity in the conduit to be in the range 1–10m s�1; thus, the diameter of a singleconduit will be in the range 30–90m. The corresponding flow Reynolds number is thus 107–108; under thesefully turbulent flow conditions the friction coefficient is around 0.005 [Massey, 1970]. The frictional pressuredrop may then be evaluated from equation (A1); it lies in the range 250 to 105 Pa.
A conduit of 30m diameter is unlikely to be present at the start of the eruption, but may develop from anexisting subglacial drainage channel by the passage of warm meltwater. We estimate the rate at such aconduit might develop by noting that 0.21 km3 of ice melted in 3 days along the conduit from the Gjálp 1996subglacial eruption site to the subglacial lake Grimsvötn, a distance of around 10 km [Gudmundsson et al.,2004]. The variation of melting rate with time during the 3 day period is not known; for the purpose of
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illustration we assume that it is constant at the average value of 810m3 s�1. The rate of change of conduitradius R with time is thus given by
dRdt
¼ 8102πRLe
(A3)
where Le is the conduit length. Integration of equation (A3) between the limits on R of 0 to 15m yields a timeof 2.5 h.
Notation
AC heat accumulation rate in the cavity, W.Cf friction factor, dimensionless.
Cpg vapor specific heat capacity, J kg�1 K�1.D diameter of conduit of circular cross section, m.d vertical dimension for convection, m.di depth of depression in ice surface, m.EF enhancement factor for heat transfer coefficient, dimensionless.G mass flow rate per unit width, kg s�1m�1.Gi thickness of ice above bedrock, m.g acceleration due to gravity, m s�2.h ice thickness, m.H cavity height, m.Hc cavity heat content, J.hf gas-vapor film heat transfer coefficient, Wm�2 K�1.hg saturated vapor specific enthalpy, kJ kg�1.hl saturated liquid specific enthalpy, kJ kg�1.kf fluid thermal conductivity, Wm�1 K�1.kl liquid thermal conductivity, Wm�1 K�1.L depth of liquid water in subglacial cavity, m.Le length of conduit, m.mg mass of steam in cavity, kg.ml mass of liquid water in cavity, kg.Nu Nusselt number, dimensionless.P pressure, Pa.PB pressure at top of subglacial cavity, Pa.PC pressure at bottom of subglacial cavity, Pa.PD static pressure where the meltwater enters subglacial lake, Pa.
Psteam steam pressure in the subglacial cavity, Pa.Pr Prandtl number, dimensionless.q fluid flow rate, m3 s�1.Q heat rate, J s�1.R conduit radius, m.
Ra Rayleigh number, dimensionless.Re Reynolds number, dimensionless.Ts cavity temperature, K.Tso steady state cavity temperature, K.t time, s.U heat transfer coefficient, Wm�2 K�1.v fluid velocity, m s�1.z height of base of subglacial cavity above bottom of subglacial lake, m.α coefficient of thermal expansion, K�1.β ratio of cavity half width to height, dimensionless.
ΔPf frictional pressure loss, Pa.
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ΔT temperature difference, K.κ thermal diffusivity, m2 s�1.λf fluid latent heat of fusion, J kg�1,λv fluid latent heat of vaporization, J kg�1.μl liquid dynamic viscosity, Pa s.ν kinematic viscosity, m2 s�1.ρ fluid density, kgm�3.ρg vapor density, kgm�3.ρi ice density, kgm�3.ρl liquid density, kgm�3.τ underpressure, Pa.θ angle of inclination of surface to horizontal, degree.
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